EEL4515 Homework - Chapter 1A PDF

EEL4515 Homework – Chapter 1A Note: You must show all work for full credit. Just correct answers will receive ½ credit. 1. Determine whether the following functions are energy signals, power signals or neither. Justify your answers. a. 𝑥𝑥(𝑡𝑡) = 1.7sin (2𝜋𝜋35...

EEL4515 Homework – Chapter 1A Note: You must show all work for full credit. Just correct answers will receive ½ credit. 1. Determine whether the following functions are energy signals, power signals or neither. Justify your answers. a. 𝑥𝑥(𝑡𝑡) = 1.7sin (2𝜋𝜋350𝑡𝑡 + 0.5𝜋𝜋) b. 𝑦𝑦(𝑡𝑡) = 17𝑒𝑒 −0.5|𝑡𝑡| c. 𝑤𝑤(𝑡𝑡) = 3 cos(2𝜋𝜋700𝑡𝑡) ∗ rect(15𝑡𝑡) 𝑡𝑡−7𝑖𝑖 d. 𝑧𝑧(𝑡𝑡) = 3 ∑∞ 𝑖𝑖=−∞ rect 2 2. Use the properties of impulse functions to evaluate (calculate or reduce) the following. a. 5𝛿𝛿 (𝑡𝑡 − 3) × 𝑡𝑡 3 b. 3cos (2𝜋𝜋20𝑡𝑡) × 𝛿𝛿(𝑡𝑡 −.35) ∞ c. ∫−∞ 7𝑡𝑡 2 𝛿𝛿 (𝑡𝑡 + 2)𝑑𝑑𝑑𝑑 ∞ 𝑡𝑡 d. ∫−∞ 7rect × 𝛿𝛿(𝑡𝑡 − 5)𝑑𝑑𝑑𝑑 8 3. Using Fourier Transforms and frequency domain allows easier analysis of Linear Time-Invariant (LTI) Systems. Let 𝑥𝑥(𝑡𝑡) = 3sin (2𝜋𝜋6𝑡𝑡 + 0.75) be the input to a LTI system. The LTI system has a transfer function 𝐻𝐻 (𝑓𝑓) = 0.7× |𝑓𝑓|. What is the output, 𝑦𝑦(𝑡𝑡), of the LTI system? 4. Circle the correct words in each of the following statements. Given that x(t) is a real function with the Fourier transform X(f): a. The real part of X(f) is an even / odd function. b. The imaginary part of X(f) is an even / odd function. 5. Using Fourier transform and properties tables, determine 𝑋𝑋(𝑓𝑓), the Fourier transform of x(t). a. 𝑥𝑥(𝑡𝑡) = 13Π(0.30𝑡𝑡) = 13rect(0.30𝑡𝑡) b. 𝑥𝑥(𝑡𝑡) = 7sinc[14(𝑡𝑡 − 3)] 6. Given the Fourier transform of x(t) is 𝑋𝑋(𝑓𝑓) = 5𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 2 (7𝑓𝑓), find the Fourier transform, Y(f), of 𝑦𝑦(𝑡𝑡) = 4𝑥𝑥(2𝑡𝑡 − 8). Do NOT determine x(t). 7. Determine 𝑋𝑋(𝑓𝑓), the Fourier Transform of 6sinc(6𝑡𝑡) × sin (2𝜋𝜋10𝑡𝑡). Hint: Use the operation #8 and the property of convolving with an impulse function. Sketch X(f).